Alternative solutions

In[35]:= Needs["DifferentialEquations`NDSolveProblems`"];
Needs["DifferentialEquations`NDSolveUtilities`"];

In[37]:= tmin = -2; tmax = 2;
 
In[38]:= (* u[x, 0] \[Equal] 4Sech[x]^2*)

In[39]:= \[Nu] = 1; \[Epsilon] = 0.001;(*0.005*);

In[40]:= xmin = -2; xmax = 2; sol = 
 NDSolve[ {D[u[x, t], t] + 
     u[x, t]*D[u[x, t], x] + \[Epsilon]^2*D[u[x, t], {x, 3}] == \[Nu]*
     D[u[x, t], {x, 2}], u[x, 0] == Cos[x], u[xmin, t] ==  u[xmax, t] }, 
  u, {x, xmin, xmax}, {t, tmin, tmax}, Method -> "Automatic"]

During evaluation of In[40]:= NDSolve::mxsst: Using maximum number of grid points 10000 allowed by the MaxPoints or MinStepSize options for independent variable x.

During evaluation of In[40]:= NDSolve::ndsz: At t == -5.72262*10^-7, step size is effectively zero; singularity or stiff system suspected.

During evaluation of In[40]:= NDSolve::eerr: Warning: scaled local spatial error estimate of 5.058814843192396`*^10 at t = -5.72262*10^-7 in the direction of independent variable x is much greater than the prescribed error tolerance. Grid spacing with 10001 points may be too large to achieve the desired accuracy or precision. A singularity may have formed or a smaller grid spacing can be specified using the MaxStepSize or MinPoints method options.

Out[40]= {{u ->                                                     -7
   InterpolatingFunction[{{\[Ellipsis], -2., 
      2., \[Ellipsis]}, {-5.72262 10  , 2.}}, <>]}}

Solve Burgers in form \[Nu] \[PartialD]u^2/\[PartialD]^2x= -(\[PartialD]u/\[PartialD]t)- u \[PartialD]u/\[PartialD]x-\[Epsilon]^2 \[PartialD]^3u/\[PartialD]x^3 

In[41]:= Plot3D[u[x, t] /. Flatten[sol], {x, xmin, xmax}, {t, tmin, tmax}, 
 PlotPoints -> 100, PlotRange -> All, ColorFunction -> "Rainbow"]

Needs["DifferentialEquations`NDSolveProblems`"];
Needs["DifferentialEquations`NDSolveUtilities`"];

tmin = -2; tmax = 2;
(* u[x, 0] \[Equal] 4Sech[x]^2*)

ν = 1; ϵ = 0.001;(*0.005*);

xmin = -2; xmax = 2;  
sol = 
 NDSolve[ {D[u[x, t], t] + 
     u[x, t]*D[u[x, t], x] + ϵ^2*D[u[x, t], {x, 3}] == ν*
     D[u[x, t], {x, 2}], u[x, 0] == Cos[x], u[xmin, t] ==  u[xmax, t] }, 
  u, {x, xmin, xmax}, {t, tmin, tmax}, Method -> "Automatic"]

During evaluation of In[40]:= NDSolve::mxsst: Using maximum number of grid points 10000 allowed by the MaxPoints or MinStepSize options for independent variable x.

During evaluation of In[40]:= NDSolve::ndsz: At t == -5.72262*10^-7, step size is effectively zero; singularity or stiff system suspected.

During evaluation of In[40]:= NDSolve::eerr: Warning: scaled local spatial error estimate of 5.058814843192396`^10 at t = -5.7226210^-7 in the direction of independent variable x is much greater than the prescribed error tolerance. Grid spacing with 10001 points may be too large to achieve the desired accuracy or precision. A singularity may have formed or a smaller grid spacing can be specified using the MaxStepSize or MinPoints method options.

(* Out[40]= {{u -> -7 InterpolatingFunction[{{…, -2., 2., …}, {-5.72262 10 , 2.}}, <>]}} *)

Solve Burgers in form ν ∂u^2/∂^2x= -(∂u/∂t)- u ∂u/∂x-ϵ^2 ∂^3u/∂x^3

Plot3D[u[x, t] /. Flatten[sol], {x, xmin, xmax}, {t, tmin, tmax}, 
 PlotPoints -> 100, PlotRange -> All, ColorFunction -> "Rainbow"]